A solved problem from Griffiths creats problem

A solved problem from Griffiths creats problem.It requires some discussion...It asks to find vector potential of an infinite solenoid with n turns per unit length radius R and current I.
Griffiths correlates integral form of Ampere's law to [closed integral of A.dl=surface integarl B.da].I can find A from this, with B=(mu)nI,uniform axial field and A circumferential.
But Griffiths says something here which I cannot understand.Quoting him,"The present problem (with a uniform magnetic field (mu)nI inside the solenoid and no field outside) is analogous to the Ampere's law problem of a fat wire carrying a uniformly distributed current.The vector potential is 'circumferential' (it mimics the magnetic field of the wire)."
As it appears, Griffiths is talking about the case where a fat wire is carrying a line current.In this case, magnetiv field is circumferential...But what is its justification?

Try solving the problem for starts. What's the magnetic field for such a wire? Hint: you can still use Ampere's law for it.

Also, it would be very helpful if you took at look at https://www.physicsforums.com/showthread.php?t=8997 It makes reading posts with math infinitely easier, and if you're going to be a physicist, you're going to have to learn LaTeX sooner or later, anyway.